In ``Representations of algebraic groups'', Pure Appl. Math. 131 (1987; Zbl 0654.20039), \textit{J. C. Jantzen} constructed certain filtrations of the Weyl modules and investigated their role in the representation theory of the reductive algebraic group \(G(\mathbb{F}_ p)\). The authors give an algebraic geometry interpretation of these filtrations: Let \(Y\) be the hypersurface defined by the equation \(X_ 0^{q+1}+X_ 1^{q+1}+\cdots+X_ n^{q+1}=0\), \(q=p^ \ell\), in \(\mathbb{P}^ n\), \(n\geq 2\). The unitary group \(G\) of \((n+1)\times(n+1)\)- matrices over the field of \(q^ 2\) elements (also its subgroup \(H\) of diagonal matrices) operates on this variety. Hence there is an induced action on the cohomology \(H^{n-1}_{crys}(Y_ k/k)\cong H_{DR}^{n-1}(Y/W(k))\). Under this action the Hodge filtration \({\mathcal F}_ r^ \bullet H_{DR}^{n-1}(Y/W(k))\) is preserved and gives the desired filtration from representation theory.